×

zbMATH — the first resource for mathematics

Development of three dimensional constitutive theories based on lower dimensional experimental data. (English) Zbl 1200.76005
Summary: Most three dimensional constitutive relations that have been developed to describe the behavior of bodies are correlated with one dimensional and two dimensional experiments. What is usually lost sight of is the fact that infinity of such three dimensional models may be able to explain experiments that are lower dimensional. Recently, the notion of maximization of the rate of entropy production has been used to obtain constitutive relations based on the choice of the stored energy and rate of entropy production, etc. In this paper, we show different choices for the manner in which the body stores energy and dissipates energy and satisfies the requirement of maximization of the rate of entropy production that can all describe the same experimental data. All of these three dimensional models, in one dimension, reduce to the model proposed by Burgers to describe the viscoelastic behavior of bodies.

MSC:
76A02 Foundations of fluid mechanics
76A05 Non-Newtonian fluids
76A10 Viscoelastic fluids
74D10 Nonlinear constitutive equations for materials with memory
PDF BibTeX Cite
Full Text: DOI EuDML arXiv
References:
[1] J. M. Burgers: Mechanical considerations-model systems-phenomenological theories of relaxation and of viscosity. First Report on Viscosity and Plasticity. Nordemann Publishing Company, New York, 1935.
[2] A. E. Green, P. M. Naghdi: On thermodynamics and the nature of the second law. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 357 (1977), 253–270.
[3] M. Itskov: On the theory of fourth-order tensors and their applications in computational mechanics. Comput. Methods Appl. Mech. Eng. 189 (2000), 419–438. · Zbl 0980.74006
[4] J. Málek, K. R. Rajagopal: A thermodynamic framework for a mixture of two liquids. Nonlinear Anal.-Real World Appl. 9 (2008), 1649–1660. · Zbl 1154.76311
[5] J. C. Maxwell: On the dynamical theory of gases. Philos. Trans. Roy. Soc. London 157 (1867), 49–88.
[6] J. Murali Krishnan, K. R. Rajagopal: Thermodynamic framework for the constitutive modeling of asphalt concrete: Theory and applications. J. Mater. Civ. Eng. 16 (2004), 155–166.
[7] J. G. Oldroyd: On the formulation of rheological equation of state. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 200 (1950), 523–591. · Zbl 1157.76305
[8] K. R. Rajagopal: Multiple configurations in continuum mechanics. Report Vol. 6. Institute for Computational and Applied Mechanics, University of Pittsburgh, Pittsburgh, 1995.
[9] K. R. Rajagopal: On implicit constitutive theories. Appl. Math. 48 (2003), 279–319. · Zbl 1099.74009
[10] K. R. Rajagopal, A. R. Srinivasa: Mechanics of the inelastic behavior of materials. Part II: Inelastic response. Int. J. Plast. 14 (1998), 969–995. · Zbl 0978.74014
[11] K. R. Rajagopal, A. R. Srinivasa: A thermodynamic framework for rate type fluid models. J. Non-Newtonian Fluid Mech. 88 (2000), 207–227. · Zbl 0960.76005
[12] K. R. Rajagopal, A. R. Srinivasa: On the thermomechanics of materials that have multiple natural configurations. Part I: Viscoelasticity and classical plasticity. Z. Angew. Math. Phys. 55 (2004), 861–893. · Zbl 1180.74006
[13] K. R. Rajagopal, A. R. Srinivasa: On the thermomechanics of materials that have multiple natural configurations. Part II: Twinning and solid to solid phase transformation. Z. Angew. Math. Phys. 55 (2004), 1074–1093. · Zbl 1125.74303
[14] K. R. Rajagopal, A. R. Srinivasa: On thermomechanical restrictions of continua. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460 (2004), 631–651. · Zbl 1041.74002
[15] K. R. Rajagopal, A. R. Srinivasa: On the thermodynamics of fluids defined by implicit constitutive relations. Z. Angew. Math. Phys. 59 (2008), 715–729. · Zbl 1149.76007
[16] I. J. Rao, K. R. Rajagopal: On a new interpretation of the classical Maxwell model. Mech. Res. Comm. 34 (2007), 509–514. · Zbl 1192.74058
[17] H. Ziegler: Some extremum principles in irreversible thermodynamics. In: Progress in Solid Mechanics, Vol. 4 (I. N. Sneddon, R. Hill, eds.). North Holland, New York, 1963.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.